## Thursday, September 13, 2018

### Irreversible time: illusion, or simplification?

 Ilya Prigogine
We know Einstein believed that the passage of time is an illusion. In a letter of condolence he wrote in 1955 he said: ...the distinction between past, present and future is only a stubbornly persistent illusion. To assert this, he relied on the fact that Newton’s equations of gravitation, his own equations of General Relativity, Maxwell’s equations (which apply to electromagnetic waves) and SchrÃ¶dinger’s equation (which gives the wave function of a particle in quantum mechanics) are all symmetric with respect to time.
How then can we explain the fact that it seems so obvious that time goes from the past to the future? Usually, physicists who believe that time is an illusion explain it by saying that, at the microscopic level, time is actually reversible, but when we move to the macroscopic level, new, emerging phenomena appear, one of which is the irreversibility of time. Let's give an example:
According to the usual theories, the movement of the molecules of a gas is perfectly reversible. If we reverse the direction of time, all the particles behave exactly the same and continue colliding with each other, only they would move in the opposite direction. However, when we consider all the trillions of particles that make up a gas, we see irreversible phenomena arising, such as the fact that the gas always tends to occupy as much space as possible, while its accumulation in a corner of the container is much less likely.
The problem is that our physical theories are based on approximations. Mathematics is a very important tool for physics, but in mathematics there are several kinds of very different problems, which differ in their difficulty to be solved. Let us look at a few:

• Linear and non-linear systems: the first, much simpler, can be described by linear equations with the following property:

• Integrable and non-integrable equations: the first are differential equations that can be solved into analytical solutions, expressible by means of relatively simple expressions. The latter cannot be integrated analytically, only approximately, by numerical calculation. For example, Einstein’s cosmological equation of General Relativity, which we have seen in several previous posts, can be easily integrated if either its second or its third term is equal to zero, but not if both are different from zero.
• Hilbert spaces: where you can define an internal scalar product whose norm, defined below, is complete.
To simplify the calculations, physicists use to prefer linear to non-linear equations; integrable to non-integrable equations; and Hilbert spaces (which are used in the mathematical formulation of quantum mechanics) to spaces not in that class.
Given this situation, the Nobel Prize winner Ilya Prigogine (1917-2003), specialist in dealing with complex dynamic systems far from the equilibrium (such as living systems), raised the following question:
Perhaps the apparent reversibility of time at the microscopic level is just a consequence of the simplifications we make in mathematical physics, rather than a real phenomenon.
 Albert Einstein
Put in another way: is time an illusion, as Einstein believed, or is time really irreversible, and the illusion of its reversibility is due to the simplification introduced by our mathematical representations?
To answer this question, Prigogine and his team investigated what would happen if quantum mechanics (the SchrÃ¶dinger equation) were extended to spaces more complex than Hilbert space. The result was that time became irreversible. Something similar happened when classical mechanics and General Relativity were tackled with non-integrable equations. Finally, even statistical mechanics (the study of the behavior of gases at the microscopic level) showed a unidirectional time when the equations were made more complicated. The solutions found, rather than representing wave functions (for example) turned into intrinsic probability distributions.
Prigogine comes to the following conclusion: saying that time is an illusion, is a simplification introduced by our equations. Let us look at his own words:
The inclusion of irreversibility changes our view of nature. The future is no longer given. Our world is a world of continuous “construction” ruled by probabilistic laws and no longer a kind of automaton. We go from a world of "being" to a world of "becoming". Nevertheless, why nature has a broken time symmetry is a difficult question. It may be due to the interaction between gravitation and the other fundamental forces. But at this point we are at the frontiers of present science. (Prigogine & Antoniou, Laws of Nature and Time Symmetry Breaking, in Tempos in Science and Nature, Annals of the New York Academy of Sciences, Vol. 879, 1999).
Prigogine wrote these words almost 20 years ago. Some of his references are even older. Has there been a debate about this question? Or has the law of silence acted here, because these ideas do not agree with the dominant scientific ideology?

The same post in Spanish
Thematic thread on Time: Preceding Next
Manuel Alfonseca